Half Life Formula Calculator: Solve Decay Easily

This calculator works out the remaining quantity of a substance after a given time using the standard exponential decay formula, N(t) = N₀ × (1/2)^(t/t½). You enter the initial amount, the half-life, and the elapsed time, and you get back the remaining amount and the fraction left. It is the same math used for radioactive decay, drug elimination, and any process that halves at a constant rate.

Formula

The half-life formula is written as:

N(t) = N₀ × (1/2)^(t / t½)

• N(t) – the amount remaining after time t (same unit as Nâ‚€, e.g., grams, milligrams, or number of atoms).
• Nâ‚€ – the initial amount at time zero.
• t – the elapsed time (any unit, as long as it matches t½).
• t½ – the half-life, the time it takes for the amount to reduce to half its starting value.

The exponent t / t½ is a dimensionless number that tells you how many half-lives have passed. If that number is 2, the substance has gone through two half-lives, leaving (1/2)² = 1/4 of the original.

Worked Example

Using the default values from the calculator above: N₀ = 100 g, t½ = 10 hours, t = 25 hours.

1. Compute the number of half-lives: 25 / 10 = 2.5.
2. Raise 0.5 to that power: 0.5^2.5 ≈ 0.1767767.
3. Multiply by initial amount: 100 × 0.1767767 ≈ 17.6777 g.

The calculator shows 17.6777 g (rounded to 4 decimals) and a fraction left of 0.1768. The same result is obtained whether you use grams, kilograms, or any other mass unit, as long as Nâ‚€ and N(t) share the same unit.

When to Use the Half-Life Formula

The formula applies to any process that follows first-order kinetics, meaning the rate of decay is proportional to the amount present. Common real-world applications include:

• Radioactive decay – carbon-14 dating, medical isotopes, nuclear waste.
• Pharmacokinetics – how long a drug stays in the body (e.g., a drug with a 24-hour half-life is mostly gone after a few days).
• Environmental science – breakdown of pollutants or pesticides in soil.
• Material fatigue – some physical properties degrade exponentially.

The formula is not for processes that decay linearly (e.g., a leaky bucket losing water at a constant rate) or for reactions that depend on the concentration of two different substances (second-order kinetics).

How to Read the Result

The first output, remaining amount, is the quantity left after the elapsed time. The second output, fraction remaining, is the proportion left relative to the start (always between 0 and 1 for positive times). For example, a fraction of 0.25 means 25% of the original remains. If you need the amount that has decayed, subtract the remaining amount from the initial amount: decayed = N₀ − N(t).

In radioactive dating, the fraction remaining is what you measure (e.g., a sample with 12.5% of its original carbon-14 has gone through 3 half-lives). The formula can also be rearranged to solve for time or half-life if you know the other values.

Common Mistakes and Pitfalls

• Unit mismatch – The most frequent error: using hours for the half-life and minutes for the elapsed time. Always convert to the same unit before calculating.
• Confusing half-life with doubling time – Doubling time uses a base of 2, not 0.5. The formula for doubling is N(t) = Nâ‚€ × 2^(t/Tâ‚‚).
• Rounding too early – If you round the exponent to 2.5 and then round 0.5^2.5 to 0.18, you lose precision. Keep extra digits until the final step.
• Using the natural exponential form incorrectly – Some people write N(t) = Nâ‚€ e^(−λt) where λ = ln(2)/t½. That is correct, but the decay constant λ is not the same as the half-life; it has units of 1/time.

Limits of the Formula

The half-life formula assumes a constant half-life, which is true for most radioactive isotopes and many chemical reactions under fixed conditions. However, it does not apply when:

• The process is not first-order (e.g., some biological elimination pathways become saturated).
• The half-life changes with temperature, pressure, or concentration (common in chemical reactions).
• The substance is being produced or replenished while it decays (e.g., a patient receiving a continuous drug infusion). In that case, you need a more complex model that includes an input term.

Also, the formula gives the expected average behavior. Actual decay is random at the atomic level, but for large numbers of atoms, the statistical fluctuation is negligible.

Frequently Asked Questions

How do I calculate half-life if I only know the initial and remaining amounts?

Rearrange the formula to solve for t½: t½ = t / log₂(N₀ / N(t)). For example, if you start with 100 g and after 5 hours you have 25 g, then N₀/N(t) = 4, log₂(4) = 2, so t½ = 5/2 = 2.5 hours.

What is the difference between half-life and decay constant?

The decay constant (λ) is the probability per unit time that a single atom will decay. It is related to half-life by λ = ln(2) / t½. The half-life is more intuitive: it is the time for half the sample to decay. Both describe the same exponential process.

Can I use this calculator for carbon dating?

Yes, if you know the initial amount of carbon-14 (usually assumed to be the same as in living organisms) and the half-life of carbon-14 (about 5,730 years). Enter those values along with the measured remaining amount to find the age. However, real carbon dating requires calibration curves, so the result is approximate.

What does a half-life of zero mean?

A half-life of zero would mean the substance decays instantly, which is not physically meaningful. The calculator requires a positive half-life. In practice, half-lives range from fractions of a second (e.g., some nuclear isomers) to billions of years (e.g., uranium-238).

Does the half-life formula work for doubling populations?

No, that is a different formula. For exponential growth, you use N(t) = N₀ × 2^(t/T₂), where T₂ is the doubling time. The half-life formula is specifically for decay.

Why is the exponent (t / t½) not a whole number?

Because time can be any value, not just whole multiples of the half-life. For example, after 1.5 half-lives, the fraction remaining is 0.5^1.5 ≈ 0.3536, which is between 0.5 (after one half-life) and 0.25 (after two). The formula handles fractional exponents smoothly.

Related

• Change in K Formula: Delta K Explained
• Change in Quantity Demanded Calculator
• kg to Pounds Calculator: Convert Kilograms

in N0 | Initial amount | g | 100 | 0
in t_half | Half-life | h | 10 | 0
in t_elapsed | Elapsed time | h | 25 | 0
out N_remaining | Remaining amount | g | N0 * (0.5)^(t_elapsed / t_half) | 4
out fraction_left | Fraction remaining | - | (0.5)^(t_elapsed / t_half) | 4